nerve and realization


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Under rather general conditions a functor

S C:SC S_C : S \to C

into a cocomplete category CC (possibly a VV-enriched category with VV some complete symmetric monoidal category) induces a pair of adjoint functors

CN||[S op,V], C \stackrel{\xleftarrow{|-|}}{\underset{N}{\rightarrow}} [S^{op}, V],

where ||N|-| \dashv N, between CC and the category of presheaves PSh(C)=[S op,V]PSh(C) = [S^{op}, V] on SS (here VV = Set for the unenriched case)



Here “SS” is supposed to be suggestive of a category of certain “geometric Shapes”. The canonical example is S=ΔS = \Delta, the simplex category, and the reader may find it helpful to keep that example in mind.


We place ourselves in the context of VV-enriched category theory. The reader wishing to stick to the ordinary notions in locally small categories takes VV= Set.

The realization operation is the left Kan extension of S C:SCS_C : S \to C along the Yoneda embedding S[S op,V]S \hookrightarrow [S^{op},V] (i.e. the Yoneda extension):

S S C C Y || [S op,V]. \array{ S &\stackrel{S_C}{\to}&& C \\ \downarrow^{Y} &\Downarrow& \nearrow_{|-|} \\ [S^{op},V] } \,.

If we assume that CC is tensored over VV, then by the general coend formula for left Kan extension we find that for X[S op,V]X \in [S^{op}, V] we have

|X| sSS C(s)X s. |X| \simeq \int^{s \in S} S_C(s) \cdot X_s \,.

For instance when S=ΔS = \Delta is the simplex category this reads more recognizably

|X| [n]ΔΔ C[n]X n. |X| \simeq \int^{[n] \in \Delta} \Delta_C[n] \cdot X_n \,.

The corresponding nerve operation

N:C[S op,V] N : C \stackrel{}{\to} [S^{op},V]

is given by

N(c):S opS C opC opC(,c)V. N(c) : S^{op} \stackrel{S_C^{op}}{\to} C^{op} \stackrel{C(-,c)}{\to} V \,.
Theorem (Kan)

Nerve and realization are a pair of adjoint functors

(||N) (|-| \dashv N)

with NN right adjoint.


Using the fact that the Hom in its first argument sends coends to ends and then using the definition of tensoring over VV, we check the hom-isomorphism

Hom C(|X|,c) :=Hom C( sS C(s)X s,c) sHom C(S C(s)X s,c) sHom V(X s,C(S C(s),c)) = sHom V(X s,N(c) s) Hom [S op,V](X,N(c)), \begin{aligned} Hom_C(|X|, c) &:= Hom_C( \int^{s} S_C(s) \cdot X_s, c) \\ & \simeq \int_{s} Hom_C( S_C(s) \cdot X_s, c) \\ & \simeq \int_{s} Hom_V( X_s , C(S_C(s), c)) \\ & = \int_{s} Hom_V( X_s , N(c)_s) \\ & \simeq Hom_{[S^{op},V]}(X, N(c)) \,, \end{aligned}

where in the last step we used the definition of the enriched functor category in terms of an end.


In many cases we have V=V = Set and the tensoring of an object cc over a set II is given by coproducts as

cI= iIc. c \cdot I = \coprod_{i \in I} c \,.

This is the case for instance for the below examples of realization of simplicial sets, nerves of categories and the Dold-Kan correspondence.


Topological realization of simplicial sets

A classical example is given by the cosimplicial topological space

Δ Top:ΔTop \Delta_{Top} : \Delta \to Top

that sends the abstract nn-simplex [n][n] to the standard topological nn-simplex Δ Top[n] n\Delta_{Top}[n] \subset \mathbb{R}^n.

This topological nerve and realization adjunction plays a central role as a presentation of the Quillen equivalence between the model structure on simplicial sets and the model structure on topological spaces. This is discussed in detail at homotopy hypothesis.

Nerve and realization of categories

Pretty much every notion of category and higher category comes, or should come, with its canonical notion of simplicial nerve, induced from a functor

Δ C:ΔnCat \Delta_C : \Delta \to n Cat

that sends the standard nn-simplex to something like the free nn-category on the nn-directed graph underlying that simplex.

For ordinary categories see the discussion at nerve and at geometric realization of categories.

One formalization of this for n=n = \infty in the context of strict ω-categories is the cosimplicial ω\omega-category called the orientals

Δ ω:ΔωCat. \Delta_{\omega} : \Delta \to \omega Cat \,.
  • The induced nerve is the ω-nerve.

  • The induced realization operation is the operation of forming the free ω\omega-category on a simplicial set. See oriental for more details.

Dold–Kan correspondence

The Dold–Kan correspondence is the nerve/realization adjunction for the homology functor

Δ C :ΔCh + \Delta_{C_\bullet} : \Delta \to Ch_+

to the category of chain complexes of abelian groups, which sends the standard nn-simplex to its homology chain complex, more precisely to its normalized Moore complex.

Simplicial models for (,1)(\infty,1)-categories

The canonical cosimplicial simplicially enriched category

ΔSSet-Cat \Delta \to SSet\text{-}Cat

induces the homotopy coherent nerve of SSet-enriched categories and establishes the relation between the quasi-category and the simplicially enriched model for (infinity,1)-categories. See


Full and faithfulness

Under some conditions one can characterize when and where the nerve construction is a full and faithful functor. For the moment see for instance monad with arities.


The notion of nerve and realization (not with these names yet) was introduced and proven to be an adjunction in section 3 of

  • Daniel Kan, Functors involving c.s.s complexes, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330–346 (jstor).

In fact, in that very article apparently what is now called Kan extension is first discussed.

Also, in that article, as an example of the general mechanism, also the Dold–Kan correspondence was found and discussed, independently of the work by Dold and Puppe shortly before, who used a much less general-nonsense approach.

In an article in 1984, Dwyer and Kan look at this ‘nerve-realization’ context from a different viewpoint, using the term ‘singular functor’ where the above has used ‘nerve’. Their motivation example is that in which SS is the orbit category of a group GG, and the realisation starts with a functor on that category with values in spaces and returns a GG-space:

  • W. G. Dwyer and D. M. Kan, Singular functors and realization functors, Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147 – 153.

We should also mention the treatment in Leinster’s book and the relation to the notions of dense subcategory or adequate subcategory in the sense of Isbell.

In a blog post on the n-Category Café, Tom Leinster illustrates that “sections of a bundle” is a nerve operation, and its corresponding geometric realization is the construction of the étalé space of a presheaf.

Revised on February 15, 2017 05:40:09 by Urs Schreiber (